Now pick an element of $\mathbb Z_{12}$ that is not a generator, say $2$. Calculate all of the elements in $\langle2 \rangle$. This is a subgroup. Repeat this for a different non-generating element. You should find $6$ subgroups.

I have got trouble in understanding the line "$\mathbb Z_{12}$ has $\phi (12)=4$ generators: $1, 5, 7$ and $11$, ..".How did I get the generators $1, 5, 7$ and $11$.Can someone explain,please?
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learnerMay 22 '13 at 11:05

$\phi(n)$ is the Euler totient function - the number of coprime numbers less than n. 1,5,7,11 are all coprime with 12 and less than 12 and are the only such numbers, thus $\phi(12)=4$. Because they are all coprime with 12 then they will generate $Z_{12}$.
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alex-1729'May 6 '14 at 18:43